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Properties

Label 16.8.13753742182...3136.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{12}\cdot 41^{14}\cdot 97^{4}$
Root discriminant $136.04$
Ramified primes $2, 41, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4.C_2^2:D_4$ (as 16T209)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-732956425, 2139886400, -226364881, -52518582, 283303175, 117404282, -94085134, -18669757, 8347337, 344801, -116329, 18246, -4901, -44, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 3*x^14 - 44*x^13 - 4901*x^12 + 18246*x^11 - 116329*x^10 + 344801*x^9 + 8347337*x^8 - 18669757*x^7 - 94085134*x^6 + 117404282*x^5 + 283303175*x^4 - 52518582*x^3 - 226364881*x^2 + 2139886400*x - 732956425)
 
gp: K = bnfinit(x^16 - 3*x^15 - 3*x^14 - 44*x^13 - 4901*x^12 + 18246*x^11 - 116329*x^10 + 344801*x^9 + 8347337*x^8 - 18669757*x^7 - 94085134*x^6 + 117404282*x^5 + 283303175*x^4 - 52518582*x^3 - 226364881*x^2 + 2139886400*x - 732956425, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 3 x^{14} - 44 x^{13} - 4901 x^{12} + 18246 x^{11} - 116329 x^{10} + 344801 x^{9} + 8347337 x^{8} - 18669757 x^{7} - 94085134 x^{6} + 117404282 x^{5} + 283303175 x^{4} - 52518582 x^{3} - 226364881 x^{2} + 2139886400 x - 732956425 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13753742182201431148415117624283136=2^{12}\cdot 41^{14}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.04$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 41, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{12} - \frac{1}{10} a^{11} - \frac{1}{2} a^{10} + \frac{1}{10} a^{9} - \frac{1}{10} a^{7} - \frac{3}{10} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{3}{10} a^{2} - \frac{3}{10} a$, $\frac{1}{10} a^{14} - \frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{1}{2} a^{6} - \frac{1}{5} a^{4} + \frac{3}{10} a^{3} + \frac{2}{5} a^{2} + \frac{3}{10} a$, $\frac{1}{2268233556870017255543056937149502010741877963279459742147277995506520111770} a^{15} - \frac{22918742220353902715937351362843184891631154604273787094175329867860747303}{2268233556870017255543056937149502010741877963279459742147277995506520111770} a^{14} - \frac{45891900170371440575306104400837382206060250820812825653546387172656417607}{1134116778435008627771528468574751005370938981639729871073638997753260055885} a^{13} + \frac{23395614146910050208149125838914533361640295017118712043798069489427558215}{453646711374003451108611387429900402148375592655891948429455599101304022354} a^{12} - \frac{41839212718310585813517915544828631438600295099707661177306959731533738177}{453646711374003451108611387429900402148375592655891948429455599101304022354} a^{11} - \frac{11868580694652776196670568455139413646765238640136423494358231131395688052}{1134116778435008627771528468574751005370938981639729871073638997753260055885} a^{10} - \frac{62475484924318466933970567158165870570541364818793747012696095946889804843}{453646711374003451108611387429900402148375592655891948429455599101304022354} a^{9} - \frac{324158694648300461635061309502783794174705170246851576397516614243255122227}{1134116778435008627771528468574751005370938981639729871073638997753260055885} a^{8} + \frac{313772497879065823177477888492308258342002973593793973436718990106435514233}{2268233556870017255543056937149502010741877963279459742147277995506520111770} a^{7} - \frac{108129753641234213833513649798709347704928753833536335247268143369032834107}{1134116778435008627771528468574751005370938981639729871073638997753260055885} a^{6} + \frac{50172703886508763681157403585189904357335563254196891371120049149834795279}{2268233556870017255543056937149502010741877963279459742147277995506520111770} a^{5} + \frac{145241313576903753229239748579753327993554220071239524864604068594398967375}{453646711374003451108611387429900402148375592655891948429455599101304022354} a^{4} + \frac{146731057979758302575106513169732879438557309199056573611666605656552582209}{453646711374003451108611387429900402148375592655891948429455599101304022354} a^{3} + \frac{151207981945753979679422918643198788879601617573161270542577707142431556781}{453646711374003451108611387429900402148375592655891948429455599101304022354} a^{2} - \frac{544574576011347734329822107921050598943281136550778319265564945963985023079}{1134116778435008627771528468574751005370938981639729871073638997753260055885} a - \frac{128625805831261341640261514246538165091540175139031653411690506047579072313}{453646711374003451108611387429900402148375592655891948429455599101304022354}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 126437991657 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^2:D_4$ (as 16T209):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 4.4.13448.1, 4.4.551368.1, 8.8.304006671424.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
41Data not computed
$97$97.8.4.2$x^{8} - 912673 x^{2} + 2036173463$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$